AFAIK in case of JPA there has been no standalone spec-provided API for
quite long time with reasoning behind that that "the API is
implementation specific and should be provided by vendors/spec
implementations" - in case of EclipseLink the repo with API is one of
the repositories managed by eclipselink project.

Or is the specification and the TCK now part of EclipseLink?

I believe that it has been clearly communicated by others that specs and
TCK's as such are being/will be handled separately; these have never
been part of EclipseLink, nor managed by it.